Search for "Algebraic groups" in article titles:

1. Linear algebraic groups, arithmetic theory of: The theory that studies arithmetic properties of linear algebraic groups (cf. Linear algebraic group), defined, as a rule, over a global field. (Mathematics) [100%] 2023-12-18

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1. Algebraic algebra: An algebra with associative powers (in particular, an associative algebra) over a field in which all elements are algebraic: an element $a$ of the algebra $A$ is called algebraic over the field $F$ if the subalgebra $F$ generated by $a ... (Mathematics) [100%] 2023-10-13
2. Algebraic group: In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. (Algebraic variety with a group structure) [94%] 2023-12-19 [Algebraic groups] [Properties of groups]...
3. Algebraic group: A group $G$ provided with the structure of an algebraic variety in which the multiplication $ \mu: G\times G \to G $ and the inversion mapping $\nu: G \to G$ are regular mappings (morphisms) of algebraic varieties. An algebraic group is ... (Mathematics) [94%] 2023-12-15
4. Lie algebra, algebraic: The Lie algebra of an algebraic subgroup (see Algebraic group) of the general linear group of all automorphisms of a finite-dimensional vector space $V$ over a field $k$. If $\mathfrak g$ is an arbitrary subalgebra of the Lie algebra ... (Mathematics) [81%] 2023-10-17
5. Lie algebra of an algebraic group: l0584801.png ~/encyclopedia/old_files/data/L058/L.0508480 104 0 104 The analogue of the Lie algebra of an analytic group, which relates to the case of affine algebraic groups. As in the analytic case, the Lie algebra of an ... (Mathematics) [78%] 2023-12-15
6. Group algebra: The group algebra of a group $G$ over a field $K$ is the associative algebra over $K$ whose elements are all possible finite sums of the type $\def\a{\alpha}\def\b{\beta}\sum_{g\in G}\a_g g$, $g ... (Mathematics) [78%] 2023-12-15
7. Diagonalizable algebraic group: An affine algebraic group $ G $ that is isomorphic to a closed subgroup of an algebraic torus. Thus, $ G $ is isomorphic to a closed subgroup of a multiplicative group of all diagonal matrices of given size. (Mathematics) [77%] 2023-10-22
8. Linear algebraic group: A linear algebraic group is an algebraic group that is isomorphic to an algebraic subgroup of a general linear group. An algebraic group $G$ is linear if and only if the algebraic variety $G$ is affine, that is, isomorphic to ... (Mathematics) [77%] 2023-12-14
9. Differential algebraic group: In mathematics, a differential algebraic group is a differential algebraic variety with a compatible group structure. Differential algebraic groups were introduced by (Cassidy 1972). [77%] 2024-10-08 [Algebraic groups]
10. Semi-simple algebraic group: A semi-simple group is a connected linear algebraic group of positive dimension which contains only trivial solvable (or, equivalently, Abelian) connected closed normal subgroups. The quotient group of a connected non-solvable linear group by its radical is semi ... (Mathematics) [67%] 2023-12-18
11. Algebraic group of transformations: An algebraic group $ G $ acting regularly on an algebraic variety $ V $ . More precisely, it is a triplet $ (G,\ V,\ \tau ) $ where $ \tau : \ G \times V \rightarrow V $ ( $ \tau (g,\ x ) = gx $ ) is a morphism of algebraic varieties satisfying the conditions ... (Mathematics) [67%] 2023-10-13
12. Groups: This is a learning resource created for the School of Media Technology Social media has developed in several Internet features. One feature for common use is "Groups". [64%] 2024-01-06 [Media Technology] [Learning projects]...
13. Groups: The conception of an operation to be carried out on some object or set of objects underlies all mathematical science. Thus in elementary arithmetic there are the fundamental operations of the addition and the multiplication of integers; in algebra a ... [64%] 2022-09-02
14. Partial group algebra: In mathematics, a partial group algebra is an associative algebra related to the partial representations of a group. [63%] 2023-12-13 [Algebras] [Representation theory of groups]...
15. Group Hopf algebra: In mathematics, the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational in the theory of quantum groups. [63%] 2023-12-15 [Hopf algebras] [Quantum groups]...
16. Modular group algebra: Let $ F $ be a field and $ G $ a group. The group algebra $ FG $ is called modular if the characteristic of $ F $ is prime, say $ p $, and $ G $ contains an element of order $ p $; otherwise $ FG $ is said to be non ... (Mathematics) [63%] 2023-10-13
17. Semi-group algebra: An algebra $\Phi(S)$ over a field $\Phi$ with a basis $S$ that is at the same time a multiplicative semi-group. In particular, if $S$ is a group, one obtains a group algebra. (Mathematics) [63%] 2023-12-18
18. Rank of an algebraic group: The dimension of a Cartan subgroup of it (this dimension does not depend on the choice of the Cartan subgroup). Along with the rank of an algebraic group $ G $ one considers its semi-simple rank and reductive rank, which, by ... (Mathematics) [60%] 2023-10-19
19. Form of an algebraic group: A form of an algebraic group $G$ defined over a field $k$ is an algebraic group $G'$ defined over $k$ and isomorphic to $G$ over some extension $L$ of $k$. In this case $G'$ is called an $L/k$-form ... (Mathematics) [60%] 2023-12-19